128,950 research outputs found
Periodic Solutions of Second Order Nonlinear Difference Equations with Singular Ï•
We obtain some existence results of solutions for discrete periodic boundary value problems with singular ϕ-Laplacian operator ∇Δuk/1-κ(Δuk)2+rkuk+mk/(uk)λ=ek,   2≤k≤N-1,           u1=uN, and     Δu1=ΔuN-1 by using the upper and lower solutions method and Brouwer degree theory, where κ>0 is a constant, r=(r2,…,rN-1), m=(m2,…,mN-1), e=(e2,…,eN-1)∈RN-2, and λ>0 is a parameter. We also give some examples with singular nonlinearities to illustrate our main results
Continuous dependence on parameters for second order discrete BVP's
Using min-max inequality we investigate the existence of solutions and thier
dependence on parameters for some second order discrete boundary value problem.
The approach is based on variational methods and solutions are obtained as
saddle points to the Euler action functional.Comment: This preprint has been submitted for publicatio
Optimal Control of Convective FitzHugh-Nagumo Equation
We investigate smooth and sparse optimal control problems for convective
FitzHugh-Nagumo equation with travelling wave solutions in moving excitable
media. The cost function includes distributed space-time and terminal
observations or targets. The state and adjoint equations are discretized in
space by symmetric interior point Galerkin (SIPG) method and by backward Euler
method in time. Several numerical results are presented for the control of the
travelling waves. We also show numerically the validity of the second order
optimality conditions for the local solutions of the sparse optimal control
problem for vanishing Tikhonov regularization parameter. Further, we estimate
the distance between the discrete control and associated local optima
numerically by the help of the perturbation method and the smallest eigenvalue
of the reduced Hessian
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